Applying the hypergeometric

Sampling without replacement gives the same mean but lower variance than sampling with replacement. Here's when this matters in practice.

The finite population correction

$\text{FPC} = \Large\frac{N-n}{N-1}$

Finite Population Correction

0.91

FPC

Sample size (n): 10 of 100

199

Sampling

10%

FPC

0.909

FPC < 0.95: Use hypergeometric for accuracy

FPC = (100 - 10) / (100 - 1) = 0.909

Scenario	FPC	Interpretation
$n = 1$	≈ 1	Single draw, almost no difference
$n = N$	0	Draw everyone, no randomness left
$n \ll N$	≈ 1	Small sample, nearly independent

Rule of thumb: If $n < 0.05N$ (sample is less than 5% of population), the binomial approximation is fine.

Quality control

You're inspecting a batch of 100 items, 10 of which are defective. You test 20 items.

What do you think?

Which gives a more precise estimate of defects: with or without replacement?

Simulate batch inspections and see how detection probability depends on sample size and defect rate:

Defect Detector

100 chips total, 5 defective

Sample size: 10

130

DetectMiss

42%

58%

P(detect)

41.6%

P(miss)

58.4%

Most defects slip through with only 10 tested

Explore how the probability of accepting a batch changes as the true defect rate varies:

OC Curve

Batch size (N): 100

20500

Sample size (n): 10 (10%)

1100

@ 5% defects

42%

@ 10% defects

67%

@ 20% defects

90%

Polling

What do you think?

Surveying 1,000 from 100,000 voters vs. 1,000 from 5,000 voters. Which needs the hypergeometric?

Card games

Dealing cards is always without replacement.

What do you think?

What's the probability of exactly 2 Aces in a 5-card poker hand?

The calculation: $P(X=2) = \Large\frac{\binom{4}{2}\binom{48}{3}}{\binom{52}{5}} = \frac{6 \times 17296}{2598960} \normalsize\approx 0.040$

When to use each

Use Binomial when...	Use Hypergeometric when...
Sampling with replacement	Sampling without replacement
Population is very large	Population is finite and small
$n < 0.05N$ (approximation)	$n \geq 0.05N$
Trials are independent	Trials are dependent

Card game draws: which distribution?

Sampling 50 from 10,000: is binomial OK? (yes/no)

Common mistakes

What do you think?

Without replacement gives lower ___?

Summary

Distribution	Mean	Variance
Binomial	$np$	$np(1-p)$
Hypergeometric	$np$	$np(1-p) \times \text{FPC}$

How you sample changes the variance, not the mean. Without replacement reduces uncertainty because outcomes become negatively correlated.

Urn: 100 balls, 30 red. Draw 20 without replacement. E[reds] = ? (whole number)

Which has HIGHER variance: binomial or hypergeometric?

What's next

We've seen that random variables can be independent (binomial) or dependent (hypergeometric). But what exactly does independence mean for random variables? That's our next topic.